LAPACK  3.10.1
LAPACK: Linear Algebra PACKage

◆ zunbdb2()

subroutine zunbdb2 ( integer  M,
integer  P,
integer  Q,
complex*16, dimension(ldx11,*)  X11,
integer  LDX11,
complex*16, dimension(ldx21,*)  X21,
integer  LDX21,
double precision, dimension(*)  THETA,
double precision, dimension(*)  PHI,
complex*16, dimension(*)  TAUP1,
complex*16, dimension(*)  TAUP2,
complex*16, dimension(*)  TAUQ1,
complex*16, dimension(*)  WORK,
integer  LWORK,
integer  INFO 
)

ZUNBDB2

Download ZUNBDB2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 ZUNBDB2 simultaneously bidiagonalizes the blocks of a tall and skinny
 matrix X with orthonomal columns:

                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]

 X11 is P-by-Q, and X21 is (M-P)-by-Q. P must be no larger than M-P,
 Q, or M-Q. Routines ZUNBDB1, ZUNBDB3, and ZUNBDB4 handle cases in
 which P is not the minimum dimension.

 The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
 and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
 Householder vectors.

 B11 and B12 are P-by-P bidiagonal matrices represented implicitly by
 angles THETA, PHI.
Parameters
[in]M
          M is INTEGER
           The number of rows X11 plus the number of rows in X21.
[in]P
          P is INTEGER
           The number of rows in X11. 0 <= P <= min(M-P,Q,M-Q).
[in]Q
          Q is INTEGER
           The number of columns in X11 and X21. 0 <= Q <= M.
[in,out]X11
          X11 is COMPLEX*16 array, dimension (LDX11,Q)
           On entry, the top block of the matrix X to be reduced. On
           exit, the columns of tril(X11) specify reflectors for P1 and
           the rows of triu(X11,1) specify reflectors for Q1.
[in]LDX11
          LDX11 is INTEGER
           The leading dimension of X11. LDX11 >= P.
[in,out]X21
          X21 is COMPLEX*16 array, dimension (LDX21,Q)
           On entry, the bottom block of the matrix X to be reduced. On
           exit, the columns of tril(X21) specify reflectors for P2.
[in]LDX21
          LDX21 is INTEGER
           The leading dimension of X21. LDX21 >= M-P.
[out]THETA
          THETA is DOUBLE PRECISION array, dimension (Q)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.
[out]PHI
          PHI is DOUBLE PRECISION array, dimension (Q-1)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.
[out]TAUP1
          TAUP1 is COMPLEX*16 array, dimension (P)
           The scalar factors of the elementary reflectors that define
           P1.
[out]TAUP2
          TAUP2 is COMPLEX*16 array, dimension (M-P)
           The scalar factors of the elementary reflectors that define
           P2.
[out]TAUQ1
          TAUQ1 is COMPLEX*16 array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           Q1.
[out]WORK
          WORK is COMPLEX*16 array, dimension (LWORK)
[in]LWORK
          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= M-Q.

           If LWORK = -1, then a workspace query is assumed; the routine
           only calculates the optimal size of the WORK array, returns
           this value as the first entry of the WORK array, and no error
           message related to LWORK is issued by XERBLA.
[out]INFO
          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
  The upper-bidiagonal blocks B11, B21 are represented implicitly by
  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
  in each bidiagonal band is a product of a sine or cosine of a THETA
  with a sine or cosine of a PHI. See [1] or ZUNCSD for details.

  P1, P2, and Q1 are represented as products of elementary reflectors.
  See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR
  and ZUNGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 199 of file zunbdb2.f.

201 *
202 * -- LAPACK computational routine --
203 * -- LAPACK is a software package provided by Univ. of Tennessee, --
204 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
205 *
206 * .. Scalar Arguments ..
207  INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
208 * ..
209 * .. Array Arguments ..
210  DOUBLE PRECISION PHI(*), THETA(*)
211  COMPLEX*16 TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
212  $ X11(LDX11,*), X21(LDX21,*)
213 * ..
214 *
215 * ====================================================================
216 *
217 * .. Parameters ..
218  COMPLEX*16 NEGONE, ONE
219  parameter( negone = (-1.0d0,0.0d0),
220  $ one = (1.0d0,0.0d0) )
221 * ..
222 * .. Local Scalars ..
223  DOUBLE PRECISION C, S
224  INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
225  $ LWORKMIN, LWORKOPT
226  LOGICAL LQUERY
227 * ..
228 * .. External Subroutines ..
229  EXTERNAL zlarf, zlarfgp, zunbdb5, zdrot, zscal, zlacgv,
230  $ xerbla
231 * ..
232 * .. External Functions ..
233  DOUBLE PRECISION DZNRM2
234  EXTERNAL dznrm2
235 * ..
236 * .. Intrinsic Function ..
237  INTRINSIC atan2, cos, max, sin, sqrt
238 * ..
239 * .. Executable Statements ..
240 *
241 * Test input arguments
242 *
243  info = 0
244  lquery = lwork .EQ. -1
245 *
246  IF( m .LT. 0 ) THEN
247  info = -1
248  ELSE IF( p .LT. 0 .OR. p .GT. m-p ) THEN
249  info = -2
250  ELSE IF( q .LT. 0 .OR. q .LT. p .OR. m-q .LT. p ) THEN
251  info = -3
252  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
253  info = -5
254  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
255  info = -7
256  END IF
257 *
258 * Compute workspace
259 *
260  IF( info .EQ. 0 ) THEN
261  ilarf = 2
262  llarf = max( p-1, m-p, q-1 )
263  iorbdb5 = 2
264  lorbdb5 = q-1
265  lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
266  lworkmin = lworkopt
267  work(1) = lworkopt
268  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
269  info = -14
270  END IF
271  END IF
272  IF( info .NE. 0 ) THEN
273  CALL xerbla( 'ZUNBDB2', -info )
274  RETURN
275  ELSE IF( lquery ) THEN
276  RETURN
277  END IF
278 *
279 * Reduce rows 1, ..., P of X11 and X21
280 *
281  DO i = 1, p
282 *
283  IF( i .GT. 1 ) THEN
284  CALL zdrot( q-i+1, x11(i,i), ldx11, x21(i-1,i), ldx21, c,
285  $ s )
286  END IF
287  CALL zlacgv( q-i+1, x11(i,i), ldx11 )
288  CALL zlarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
289  c = dble( x11(i,i) )
290  x11(i,i) = one
291  CALL zlarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
292  $ x11(i+1,i), ldx11, work(ilarf) )
293  CALL zlarf( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11, tauq1(i),
294  $ x21(i,i), ldx21, work(ilarf) )
295  CALL zlacgv( q-i+1, x11(i,i), ldx11 )
296  s = sqrt( dznrm2( p-i, x11(i+1,i), 1 )**2
297  $ + dznrm2( m-p-i+1, x21(i,i), 1 )**2 )
298  theta(i) = atan2( s, c )
299 *
300  CALL zunbdb5( p-i, m-p-i+1, q-i, x11(i+1,i), 1, x21(i,i), 1,
301  $ x11(i+1,i+1), ldx11, x21(i,i+1), ldx21,
302  $ work(iorbdb5), lorbdb5, childinfo )
303  CALL zscal( p-i, negone, x11(i+1,i), 1 )
304  CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
305  IF( i .LT. p ) THEN
306  CALL zlarfgp( p-i, x11(i+1,i), x11(i+2,i), 1, taup1(i) )
307  phi(i) = atan2( dble( x11(i+1,i) ), dble( x21(i,i) ) )
308  c = cos( phi(i) )
309  s = sin( phi(i) )
310  x11(i+1,i) = one
311  CALL zlarf( 'L', p-i, q-i, x11(i+1,i), 1, dconjg(taup1(i)),
312  $ x11(i+1,i+1), ldx11, work(ilarf) )
313  END IF
314  x21(i,i) = one
315  CALL zlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, dconjg(taup2(i)),
316  $ x21(i,i+1), ldx21, work(ilarf) )
317 *
318  END DO
319 *
320 * Reduce the bottom-right portion of X21 to the identity matrix
321 *
322  DO i = p + 1, q
323  CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
324  x21(i,i) = one
325  CALL zlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, dconjg(taup2(i)),
326  $ x21(i,i+1), ldx21, work(ilarf) )
327  END DO
328 *
329  RETURN
330 *
331 * End of ZUNBDB2
332 *
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
zdrot
subroutine zdrot(N, ZX, INCX, ZY, INCY, C, S)
ZDROT
Definition: zdrot.f:98
zscal
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
zlarfgp
subroutine zlarfgp(N, ALPHA, X, INCX, TAU)
ZLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: zlarfgp.f:104
zlacgv
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:74
zlarf
subroutine zlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
ZLARF applies an elementary reflector to a general rectangular matrix.
Definition: zlarf.f:128
zunbdb5
subroutine zunbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
ZUNBDB5
Definition: zunbdb5.f:156
dznrm2
real(wp) function dznrm2(n, x, incx)
DZNRM2
Definition: dznrm2.f90:90
Here is the call graph for this function:
Here is the caller graph for this function: